The twenty-seventh post is the February 19, 2013 version in the usual place.

The early days of scheme theory

Drafts of the final two chapters are now complete.  At this point, all the mathematical material is essentially done.  The list of things to be worked on is now strongly finite (well under 150 items).   With the exception of formatting, figures, the bibliography, and the index, I am very interested in hearing of any suggestions or corrections you might have, no matter how small.

Here are the significant changes from the earlier version, in order.

The first new chapter added is the Preface.  There is no mathematical content here, but I’d appreciate your comments on it.  These notes are perhaps a little unusual, and I want the preface to get across the precise mission they are trying to accomplish, without spending too much time, and without sounding grandiose.  I’ve noticed that people who have used the notes understand well what they try to accomplish, but those who haven’t seen them are sometimes mystified.

In 10.3.9 there is a new short section on group varieties, and in particular abelian varieties are defined, and the rigidity lemma is proved.  Although it isn’t possible to give an interesting example of an abelian variety in these notes other than an elliptic curve, it seemed sensible to at least give a definition.

In 20.2.6, a short proof of the Hodge Index Theorem is given (on the convincing advice of Christian Liedtke).

And the last new chapter is Chapter 29, on completions.

29.1 is a short introduction.

29.2 gives brief algebraic background.  It concludes with one of the two tricky parts of the chapter, a theorem relating completion with exactness (and flatness).

In 29.3  we finally define various sorts of singularities.

In 29.4, the Theorem of Formal Functions is stated; this is the key result of the chapter.  Note:  the proof is hard (and deferred to the last section of the chapter).  But other than that, the rest of the chapter is surprisingly (to me) straightforward.

A formal function

In 29.5, Zariski’s Connectedness Lemma and Stein Factorization are proved.  As a sample application, we show that you can resolve curve singularities by blowing up.

In 29.6, Zariski’s Main  Theorem is proved, and some applications are given.  For example, we finally show that a morphism of locally Noetherian schemes is finite iff it is affine and proper iff it is proper and quasifinite.

In 29.7, we prove Castelnuovo’s Criterion (paying off a debt from the chapter on 27 lines), and discuss elementary transformations of ruled surfaces, and minimal surfaces.

Finally, in 29.8, the Theorem of Formal Functions is proved.  I am following Brian Conrad‘s excellent explanation (and I thank him for this, as well as for a whole lot more).  It applies in the proper setting (not just projective), and is surprisingly comprehensible; it builds on a number of themes we’ve seen before (including  Artin-Rees, and graded modules).  I think Brian told me that he was explaining Serre’s argument.  The proof is double-starred, but I hope someone tries to read it, and makes sure that I have not mutilated Brian’s exposition.

What next?

I’m going to continue to work on the many loose ends, and to fix things that people continue to catch.  There are also a number of issues on which I want to get advice (on references, notation, etc.).  I think it makes sense to ask all at once, rather than having the questions come out in dribs and drabs (as in that case people may read the first few, but then stop paying attention).  So I intend to do this in the next post, and likely within a month.

Here is an example of the sort of thing I will ask, that is relevant for the chapter just released.  There is a kind of morphism that comes up a lot, and thus deserves a name.  Suppose $\pi: X \rightarrow Y$ is a proper morphism of locally Noetherian schemes (Noetherian hypotheses  just for safety), such that the natural map $\mathcal{O}_Y \rightarrow \pi_* \mathcal{O}_X$ is an isomorphism.  Can anyone think of a great name for such a morphism?  I’d initially thought about using “Stein morphism”, but that’s terrible (as pointed out by Sándor Kovács), because it suggests something else (from complex geometry).  Sándor has suggested “connected morphism”, and Burt Totaro correctly points out that this is the “right” version of “connected fibers”, but this seems imperfect because it suggests something slightly wrong.  I think that “contraction” would be good, but that’s already used in higher-dimensional geometry (more precisely, the contractions there are these types of morphisms).  Someone (my apologies, I can’t remember who) suggested “algebraic contraction”, which seems somehow better.  But for now, I’ve not called it anything, and perhaps it is better that way.

The twenty-sixth post is the December 17, 2012 version in the usual place.  A large number of small improvements have been made, and the exposition has converged substantially, although there isn’t much big to report.

The (small) changes:

The terminology “local complete intersection” is changed to “regular embedding” (the more usual language, along with “regular immersion” — note that I have gone with “embedding” rather than “immersion” throughout).  This was because the notation I was using would cause confusion because of the existence of an importance class of morphisms called “local complete intersection morphisms” (“lci morphisms”).

I am more careful about distinguishing the canonical bundle of a smooth projective variety (the determinant of the cotangent bundle) from the dualizing sheaf, before they are identified in the last chapter, to avoid anyone being confused about what is being invoked when.

A proof of the classification of vector bundles on the projective line (sometimes known as Grothendieck’s Theorem) is given in 19.5.5.  (A proof is possible by painful algebra, and another proof is possible using Ext’s.  Because of what else is discussed in the notes, I take a middle road:  an easy proof without Ext’s, which relies on an easy calculation with 2 by 2 matrices.  Note:  I have not done the important fact that $Ext^1$ classifies extensions, which has tied my hands a little.)  This required popping out “a first glimpse of Serre duality” into its own section (19.5).

Some discussion relating to Poncelet’s Porism is added in 20.10.7.

Still to do.

The introduction and the Zariski’s Main  Theorem / Formal Functions chapter are still to be written.  (And of course things like the index, figures, and formatting won’t be dealt with until the end.)   The only other substantive things still to be written are a brief discussion of radicial morphisms (for arithmetic folks) and a brief proof and discussion of the Hodge Index Theorem (for geometric folks).  Other than that, I expect essentially no other new material to be added.

I’ve caught up with almost all of the corrections and suggestions people have given me, except for a double-digit number of pages from both Peter Johnson and Jason Ferguson.  (Thanks in particular to the comments of the reading group at Stanford for useful comments:  Macky, Brian, Zeb, Michael, Evan, and Lynnelle!)

My to-do list still has 212 things on it.  But that is a drastic improvement; the notes are converging rapidly.

I will not make any further progress until January.    I hope to have a draft of the introduction done in January, and the final substantive chapter in February.

Happy holidays everyone!

The twenty-fifth post is the October 10, 2012 version in the usual place. (Update Oct. 24: a newer version, dated October 23, 2012, is posted there now. Some of the changes are discussed in the fourth comment below.) The discussion of smooth, etale, and unramified morphisms has been moved around significantly. Johan de Jong pointed out that “unramified” should best have “locally finite type” hypotheses, thereby making its link with the other two notions more tenuous; and Peter Johnson pointed out that one could give the definition of smoothness much earlier, at the cost of initially giving an imperfect definition (a trade-off I will happily take).

I am very interested in having these changes field-tested. (Most of the rest of the notes are now quite robust thanks to the intense scrutiny they have been subjected to.) I know that when something is revised, the revisions are looked at much less. But I am hoping that someone hoping to learn about smoothness, or solidify their understanding, will give this a shot in the next couple of months. I know that in the course of doing this, my understanding of these ideas has been radically improved. Because the actual algebra was elided in most of the “standard sources”, I hadn’t realized what was important and what was unimportant, and what didn’t need to be hard and what needed to be hard. So I can at least make a promise to many readers that they might learn something new.

Here are the changes, along with suggestions of what to read (for those who have read earlier versions).

Chapter 13: Nonsingularity

13.2.8 The Smoothness-Nonsingularity Theorem is an important player. (a) If k is perfect, every nonsingular finite type k-scheme is smooth. (b) Every smooth k-scheme is nonsingular. This gets stated early, but proved late. To read: the statement of the Theorem. (To experts: Am I missing easy proofs? I think it has to be as hard as it is. Update Oct. 24, 2012: David Speyer and Peter Johnson have outlined proofs in the comments below, using just the technology of Chapter 13.)

In 13.4, I had a bad exercise, which stated that if l/k is a field extension, and X is a finite type k-scheme, then X is smooth if and only if its base change to l is smooth. One direction is easy, but I’m not even sure how to do the other direction at this point in the notes. This converse direction is now 22.2.W, which I’ll discuss bellow. To read: nothing.

13.7 is the new section on smooth morphisms, including a little motivation. Everything is easy, except showing that this definition of smooth morphisms correctly specializes to the older definition of smoothness over a field. (Notational clash that I have not resolved: the “relative dimension” of a smooth morphism is n in this section, but was d earlier. There are reasons why I couldn’t change the n to d and vice versa. I don’t think this will be confusing. (But in general I have tried hard to be consistent with notation.) To read: these 3 1/2 pages.

Chapter 22: Differentials

22.2.28-30 (a very short bit): Here a second (third?) definition of smoothness over a field is given (as we can now discuss differentials) — this was in the older version. The second definition allows us to check smoothness on any open cover, for the first time, which in turn allows us to more easily check (in 22.2.W) that smoothness of a finite type k-scheme is equivalent to smoothness after any given base field extension. This in turn allows us to establish an important fact in 22.2.X: a variety over a perfect field is smooth if and only if it is nonsingular at its closed points. This had early been in Chapter 13, but relied on 22.2.W. This also establishes part of the Smoothness-Nonsingularity Comparison Theorem. To read: 22.2.W and X (very short). Update Oct. 24, 2012: in the Oct. 23 version, this is now made into a new section, 22.3, which also includes generic smoothness. 22.2.W and X are now 22.3.C and D.

22.5: Unramified morphisms are now discussed here. This section is easy. To read: 1.5 pages. (Update October 24, 2012: the new section 22.3 bounces this section forward to 22.6 in the Oct. 23 version.)

Chapter 26: Smooth, etale, and unramified morphisms revisited

This chapter is notably shrunk. 26.1 still has motivation, but now the definitions I used to give are now just “Desired Alternate Definitions”. 26.2 now discusses “Different characterizations of smooth and etale morphisms, and their consequences”. The central (hard) result is Theorem 26.2.2, which gives a bunch of equivalent characterizations of smoothness. Before proving it, a number of applications are given. The statement of Theorem 26.2.2 is rearranged, the proof is the same. To read: skim 26.1 and read 26.2 up until the (and not including) the proof of 26.2.2: 4 light pages.

Challenge Problems:

There are some problems I would still like to see worked out by real people.
26.2.E: I moved this from the “unramified” section, but should probably move it back, as I think it can be done with what people know there.
26.2.F: Is this gettable?

The twenty-fourth post is the September 25, 2012 version in the usual place. The discussion of smoothness is now incorporated. In particular, the second-last chapter to be public is now out (Chapter 26).

As always, these changes caused ripples of changes throughout the earlier chapters. I continue to be fascinated by how intricately interconnected algebraic geometry is. There are exercises worded in a particular way early on because they come into play 20 chapters later.

As usual, I am very interested in comments, corrections, and suggestions, no matter how small. (Except: as usual, I am not yet interested in issues of formatting, figures, indexing, and other things that are best dealt with en masse later.)

Larger changes, and philosophy

Smooth morphisms are surprisingly complex. There are a number of possible definitions, and it is nontrivial to connect them all. Examples include EGA (first define formal smoothness, then add local finite presentation), Mumford’s red book (flat, geometric fibers are nonsingular, and implicitly local finite presentation), and the Stacks Project (in terms of the naive cotangent complex). I decided to take as simple a definition as possible, that I could motivate and get as much out of as cheaply as possible. (My definition: $\pi$ is said to be smooth of relative dimension $n$ if it is locally finitely presented, flat of relative dimension $n$, and $\Omega_{\pi}$ is locally free of rank $n$.) However for a number of reasons (both philosophical and because of desired consequences), I also wanted to have a clean local model of smooth morphisms (of relative dimension $n$), which is: $\text{Spec} B[x_1, ..., x_{n+k}] / (f_1, ..., f_k) \rightarrow \text{Spec} B$, where the Jacobian matrix of the $f_i$ with respect to the first $k$ of the $x_i$ is nonzero. The main difficult (important) result in the chapter is essentially this (or essentially equivalently, the connection to Mumford’s definition), Theorem 26.2.4. I found no way of making this easier. [Update Sept. 26, 2012: I forgot to mention that in the non-Noetherian case, the proof that such a map of affine schemes is flat uses a version of the local criterion for flatness that I didn’t prove. I refer to tag 046Z in the stacks project for a proof. This is yet another sign that this connection is difficult.]

(To preempt some objections: I fully agree that the definition in EGA is important, and I don’t know how to get at the left-exactness of the relative cotangent or conormal sequences otherwise. But if you want to know about it, it should be relatively short but hard work after reading this chapter. And if you don’t feel the need to know about it, there is no advantage to forcing it on you. I also think the approach of the stacks project is very nice and clean, and very possibly my preferred approach after one first meets smoothness — do not be intimidated by the phrase “cotangent complex” in this context!)

In 26.1 I motivate the definitions. In 26.2 I prove their main properties. In 26.3 I prove generic smoothness (in the source and target), and the Kleiman-Bertini theorem.

There are also a number of changes in chapter 13, on nonsingularity. If you want to revisit this chapter to see what changed, look at 13.2-4. The change with the fewest repercussions is the addition of Bertini’s theorem in 13.3. (I had originally put it in the smoothness chapter; I now realize that it is so elementary that it can be done as soon as nonsingularity is introduced.) More seriously, I realized that we can prove, and need to prove (for later exposition), some things I’d stated as facts. Most notably, we now prove what I call the Smooth-Nonsingularity Comparison Theorem, which basically says that over perfect fields, smoothness equals nonsingularity (not just over closed points), and over arbitrary fields, smoothness implies nonsingularity. So 13.2 and 13.4 are rearranged (they used to be one section). 13.4 is now called “More Sophisticated Facts”.

In particular, in previous versions I made a big deal about the fact that smoothness was important, and we really didn’t care about nonsingularity. I now realize that we do care about nonsingularity when you actually prove foundational things — for example, when we use the fact that local rings are integral domains, or “slicing” inductive arguments.

Smaller changes

• In 22.4, I define Fano, Calabi-Yau, and general type varieties, and K3 surfaces. They are easy to define, and it is good to see some examples.
• The discussion of Cohomology and Base Change is moved out of the flatness chapter into a new chapter (30) near the end, so readers don’t make the mistake of reading it before reading about smoothness.
• The chapter on the 27 lines is moved back to Chapter 27, before some of the facts needed are established. This is basically because it is morally necessary that the 27 lines chapter be chapter 27. But I will (in a later edit) encourage the reader to read this chapter before getting all the background. It is a reasonable end to a long period of learning this material, and I’d hope people jump to it as soon as they are able.

Still to come

There is only one substantive chapter left, on formal functions and related ideas (Stein factorization and Stein morphisms, Zariski’s Main Theorem, Castelnuovo’s criterion, …). The rest of the notes rely on this chapter remarkably little, so I can spend some time smoothing what is already there. I doubt I will get to this last chapter before January (after I am done teaching, and when I begin my first sabbatical). I’ll get smaller units of time to think about this, which are more suited to working through the (shrinking) to-do list.

Challenge problems

There are a number of exercises I would very much like people to try. They contain a lot of insight, and I hope they are gettable, and if they are not gettable, I want to improve them. If anyone tries these, please let me know. I am happy to give hints, and help in all ways (as I am with all exercises). The ones I am most curious about, in order of appearance, are:

• 13.2.F (“smoothness is insensitive to extension of base field”). Update October 10, 2012: as discussed in the 25th post, one of the two directions is not gettable at this point; this half has been moved.
• 22.4.J(b) (a sister to 26.2.J below)
• 26.2.G(b) (in particular, I want to be sure I’m not mistaken in thinking that local finite presentation is not needed in the argument)
• 26.2.J (especially (b), which is used in Kleiman-Bertini)
• 26.2.K (I am hoping this is straightforward)

The twenty-third post is the September 5, 2012 version in the usual place. The significant new additions deal with local complete intersections, regular sequences, depth, and Cohen-Macaulayness. In particular, the third-last chapter to be public is now out (Chapter 28). (Still to come: formal functions and related notions; and etale/smooth/unramified morphisms.)

As always, I am very interested in comments, corrections, and suggestions, no matter how small. (Except: as usual, I am not yet interested in issues of formatting, figures, indexing, and other things that are best dealt with en masse later.)

The new chapter (Chapter 28) is only 10 pages long, but there are notable changes earlier on. Moreover, you may be surprised that the file is shorter — the last page is now p. 690, not 692. This is all a sign that my understanding of these topics has greatly improved while placing them into the architecture of these notes. (Expressions of gratitude: Long ago, Karen Smith and Mike Roth separately explained how to think about these ideas in a good way. More recently, Burt Totaro helped me clean up my thinking, and pointed me to some good references.) Many of the things I wanted to add were better discussed as part of earlier discussions, rather than waiting so late. (I am conscious that this makes the earlier chapters even longer. I am aware of how foie gras is made, and I do not want to do the same thing to these notes, or to the reader.)

Here is a list of changes, culminating with the new chapter 28 on “Depth and Cohen-Macaulayness”. Some specific suggestions for learners are included, as well as some questions for other readers, especially about whether some exercises are gettable given what I say.

9.4 Regular sequences and locally complete intersections
is a new section in the chapter on closed subschemes. Regular sequences are introduced here. Effective Cartier divisors are now introduced here (rather than being some stray comment wherever they were before), as a first example. The reason for the “local” nature of regular sequences is discussed, and the key hard result is that regular sequences remain regular upon any reordering if the ring is local (Theorem 9.4.4). This section is not hard, although I fear it may be distracting. But I feel happier that local complete intersections have a key place in the exposition. (They did not come up naturally for me when I first learned algebraic geometry, and I paid a price for this.)

13.4 Regular local rings are integral domains
is a new section, whose main purpose is the theorem stated in the title. This is a surprisingly hard fact, and I now use it later several times. My goal (as always) is to get what I need as quickly as possible, rather developing all the surrounding theory. Length (used later) and associated graded rings (not used later) are introduced.

Some interesting points:

• In the appendix to Fulton’s Intersection Theory, Lemma A.6.2, gives a delightful short proof using blowing-up; it implicitly works only for varieties. Geometers may like it.
• From the inequality $d(A) \geq \rm{dim}(A)$ ($d(A)$ is the degree of the Hilbert polynomial of a local ring $A$), we can get a proof of Krull’s Principal Ideal Theorem. But I need the “multi-equation” version of Krull’s theorem, and I couldn’t see how to prove it in as easy a way, so I’ve kept the proof of Krull’s theorem currently in the notes, which is less motivated, but short and double-starred. But if anyone knows of a short proof of the multi-equation version of Krull’s theorem in this vein, please let me know!
• It is true that $d(A) = \rm{dim}(A)$, but we don’t need it, so I don’t go through the significantly extra work to show it.
• A question about something earlier in Chapter 13: I am disturbed that I know why a smooth $k$-scheme is nonsingular at closed points only if $k$ is perfect. (The statement for general $k$ is, for example, in tag 00TT of the stacks project, but I find this a depressingly hard fact.) Does anyone know of an easy explanation?

For learners: please let me know if you can understand this section, and can do the exercises. In particular, 13.4.C, 13.4.G, and 13.4.H are important — can you get them? Exercises 13.4.D and 13.4.E are lots of fun.

Excised sections:

• The old section 13.3 on “two pleasant [unproved] facts” is now cannibalized, as the theorem that localizations of regular local rings are regular is now stated back in 13.2.14 (and we now prove the case of localizations of finite type algebras over perfect fields, see Thm. 13.2.15 and the Chapter 22 discussion below), and the theorem that regular local rings are UFDs is stated here in 13.4.
• Section 13.7 on completions is now removed, as it is no longer used. (It used to be used to show that regular local rings were integral domains in the case of particularly nice varieties.) The only downside is that I’ve lost the definition of “node”, which at some point will have to be put back in.
• On a related note: I removed the section on flatness and completion (in the flatness chapter), because it never was used, and it allowed me to remove 13.7.

In Chapter 22 (Differentials):

• The conormal sheaf to a local complete intersection is easily and quickly shown to be locally free, in Proposition 22.2.16.
• The conormal exact sequence is shown to be left-exact for closed embeddings of smooth varieties in Theorem 22.2.26. I was surprised at how easy this was. (Is this similarly easy to learners? Admittedly, it uses a lot of things from earlier on.) I did not do left-exactness in more general situations, but gave references (Remark 22.2.27). It is possible I will do some of this in the forthcoming chapter on smoothness, but I very possibly will not — we don’t need it, and right now it seems like hard work.
• The fact that localizations of regular local rings are regular for localizations of finitely generated algebras over perfect fields (Theorem 13.2.15) is proved in 22.4.10 and 22.4.J. This was surprisingly easy to me. (Do you agree? Disagree?) So we finally know that affine space over a perfect field is nonsingular!
• Learners: can you get Exercise 22.3.D on differentials of discrete valuation rings? I’ve reworked it, following an excellent suggestion of John Pardon.

Chapter 23: Blowing up
This starred chapter used to be Chapter 19, but is now moved after differentials, because the conormal sheaf/cone/bundle plays an important role in later subsections. Now added: for a local complete intersection, the exceptional divisor is the projectivized normal bundle (and related facts, see Exercise 23.3.D), and the blow-up of a smooth subvariety of a smooth variety is smooth (Theorem 23.3.10). This relies on the fact that for a local complete intersection, $Sym^n(I/I^2) \rightarrow I^n/I^{n+1}$ is an isomorphism. This requires a little work; I followed Fulton’s slick argument in A.6.1 in Intersection Theory.

We finally come to:

Chapter 28: Depth and Cohen-Macaulayness.

Although Koszul complexes are a central tool for understanding depth and Cohen-Macaulayness, I avoid using them, following my usual philosophy of moving as briskly as possible to what we need, and not developing surrounding theory.

18.1 is introduces depth of Noetherian local rings. Because I find it an algebraic rather than geometric concept, we concentrate on developing some geometric sense of what it means. The main technical result in this section is the argument the cohomological interpretation of maximal sequences.

18.2 introduces Cohen-Macaulayness. A number of results are shown (e.g. equidimensionality, no embedded points, slicing criterion). Important but unneeded results are only stated in 28.2.13. A highlight of this section is the short proof of the “miracle flatness theorem”, which we make important use of in the two last chapters. (It is highly possible that this name is due to Brian Conrad. I like it.)

18.3 gives Serre’s criterion for normality. It seemed worth including, because we would like to know that regular local rings are integrally closed (with proof, not just invoking a fact we haven’t proved), so we can use all of our foundational work on normal schemes. But I’ve starred this section, in the hopes that people will read it only if they really want to.

Question for experts: In 18.3, to show that regular local rings are normal, I need the fact that they are R1. Is there an easy explanation for this fact? Currently, I quote the hard fact that localizaton of regular local rings are local, which I actually prove in the case most interesting to most people (finite type over a perfect field, see Theorem 13.2.15 above).

For learners:
Please let me know which exercises you find difficult! (And, if you have the time, even which exercises you solved!)
Exercises you should certainly try: 28.1.A, 28.1.F, 28.2.A, 28.2.B, 28.2.D, 28.2.E (the most fun exercise in this chapter), 28.2.F, 28.3.B.
Please tell me how hard you find these, and if you get them: 28.2.D and 28.2.F.

The year’s course has come to an end.  (I would like to give huge thanks to the people in the class — their detailed comments and suggestions have led to a vast number of improvements.)  A revised version is now posted at the usual place (the August 16, 2012 version).

I may later write a post with some thoughts on how the course went.  (In short: I think it showed that it is possible to cover the amount of material I wanted to, in a single year.  There were two topics I wanted to cover, but did not because I have not yet written up the exposition; but they were replaced with other useful facts.)  But right now, I want to post the latest version of the notes, with a substantially new exposition of the proof of Serre duality (the final chapter — the missing chapters are earlier) for projective varieties $X$.

I have reverted to an earlier proof I gave in versions years ago, in terms of finite (often flat) morphisms.  (I was partially prompted by an email discussion with Yuhao Huang at Berkeley — thank you Yuhao!)

There are a number of possible statements of Serre duality one might want, with the expected trade-offs:  better statements require more work.  I found productive (both personally and pedagogically) to discuss these trade-offs, and to make some decisions, and to see (in class) where approaches broke down, and where they worked.

I concentrated on several desiderata.  We want some some of duality involving a dualizing sheaf.  We will want the dualizing sheaf to be the determinant of the cotangent bundle.  This is surprisingly hard, and left to the end.

There will be potentially three versions of each desired type of duality.  There is the dimensional (lame) version, saying that the dimension of two cohomology groups are the same.  Better, there should be some duality between the cohomology groups, that should be functorial in the sheaf/bundle (“functorial Serre duality”).  And best of all, it should arise from a cup product of some sort (“trace version” of Serre duality), which requires defining the cup product.

The first kind of duality one might ask for (because we used it in discussing curves and surfaces) is Serre duality for vector bundles.  Better yet, one can have it for coherent sheaves.  (And we can get better still:  Serre duality for families; dualizing complexes; etc.  But the brutal demands of doing it all within a course, with proofs, means that we do not go there.)

In order to move toward proving these things (or even to move toward making some of the statements precise), we have to discuss Ext groups, which wasn’t hard given what we had done earlier.  As an aside (double-starred), I discussed the cup product for Ext, which of course is needed for the best statements of Serre duality.  (But it is not needed for what I prove, and what I need!)  The method of proof is to do Serre duality for projective space, and then to go to finite flat covers.  The key trick is to use a version of the “upper shriek” construction, an occasional adjoint to $\pi_*$ (even if isn’t quite “upper shriek”, so I denoted it $\pi^!_{sh}$ rather than $\pi^!$).  A complication is added by the fact that we are working with O-modules, but the construction I give only works for quasicoherent sheaves; and it works best for closed immersions.  I won’t say much more here, because it will only confuse you — it is best to read the notes.

With this approach, Serre duality (in the versions we use) ends up being surprisingly easy — it seems to come out of nowhere.  (This was independently stated to me by three people in the class, and I agree.)  But there is a surprising amount of subtlety hidden in the exposition here.  The subtlety isn’t in the arguments, or what is said; it is in the choice of what to say, and what paths to take.   There were many points where I tried to take a different route, and then something bad happened, forcing me to retract.  (The version from earlier this year proved Serre duality for projective schemes using closed immersions into projective space, but I got myself into difficulty, as observed by a number of people, including Yuhao Huang, Yuncheng Lin, Preston Wake, Charles Staat, and Yifei Zhu.  This approach ended up being much cleaner.)

Finally, this approach does not easily show that the dualizing sheaf of a smooth projective variety is the “sheaf of algebraic volume forms” (the determinant of the cotangent bundle).  To do this requires more work, and a different approach, which also allows us to prove the adjunction formula.  (However, it *is* possible to prove this directly using the “finite flat cover” approach.  Matt Baker and Janos Csirik worked it out in this note.  (I thank both Matt and Janos for permission to post this here.)

Aside:  I realized that alternative expositions can work as well, with different costs and benefits.  For example, the category of quasicoherent sheaves on a variety actually has enough injectives, so one can work directly in that category.

What comes next.  My to-do list has been finite and shrinking, but there are many things left to do.  There are a number of loose ends in the chapters already done.   I had hoped to finish the three final chapters this summer, and now I am just going hoping to finish a draft of one of them (on regular sequences).

A revised version is now posted at the usual place (the March 25, 2012 version). We have reached the end of the second quarter of our academic year, so I want to pause and look back on where we are, and fill in those who are just watching the notes evolve. (The course webpage is here.)

If we continue at the current breakneck pace, we will finish all the central material I have claimed can be covered in a single-year course. We may not succeed, but it will not be because the goal is impossible. (Instead: I have some material still to think through and prepare, and I may not manage it to my satisfaction.) I am well aware that I have 30 weeks to work with (longer than the academic year at most universities), and the people in the class are not typical, in many ways.

More precisely: in the notes, we’ve reached elliptic curves (we will begin the next quarter showing that they are group schemes). I consider everything up to 21.8 to be in very good shape. There are things that still need fixing, but I have an explicit finite list, which is large, but shrinking. I have no sections that (in my mind) need serious revision before 21.9.

Here are some rambling thoughts, both large and small, in the order in which they appear in the text. Before I begin, I should say that there are many many improvements, due to people in my class, but also a large number of sending emails from elsewhere on the globe, and also posting here. I want to repeatedly thank you for the huge number of comments you have sent in.

The section on valuative criteria (13.5) is now in potential “final form”. In other words, it is now self-contained, and open for criticism. I state the criteria (6 in total: valuative criteria for separatedness, universal closure, and properness, each in “DVR” and “general” versions), but do not prove them. I sketch the proof of the valuative criterion for separatedness in the DVR case (I basically give the proof). This is based on the discussion in the post on valuative criteria here. Please feel free to complain! (Any attempt to give a complete proof of the valuative criterion of properness ended up being longer than I wanted to include at this point.)

Fun fact (14.5.B): suppose you have a short exact sequence of quasicoherent sheaves. If the first and third are locally free, then so is the second. If the second and third are locally free and of finite rank, then so is the first. I had wondered about a counterexample if the “finite rank” hypotheses were removed. Daniel Litt has given me one, and posted it here. (Perhaps this or something like it is in the literature? Perhaps this should be added to the stacks project?)

I am mildly curious about the following (cf. 16.4). (Not curious enough that I’ve given it any thought, but curious enough that I’m hopeful someone has a very fast answer.) If $S_*$ is a graded ring, and $M_*$ is a graded $S_*$-module, if $M_*$ is finite type, is the corresponding quasicoherent sheaf finite type? And similarly for coherence? Presumably yes.  (Update June 29, 2012:  Fred Rohrer has explained this now, see below.)

The way in which I first discuss pullbacks has evolved (17.3); three different approaches all come into it (the affine-local picture; the universal property; and the “inverse image then tensor with structure sheaf” definition). (Feedback I’d earlier gotten: one expert prefers a more general approach, doing things for ringed spaces; two learners found the exercises surprisingly straightforward. So far I’m sticking with straightforward over general.)

The notion “generated by global sections” is slightly awkward, especially when relativized. I’m using the terms “globally generated” (16.3), “finitely globally generated” (16.3), and “relatively globally generated” (18.3.7). If this potentially bothers you, please complain. Ideally make a counteroffer, or at least an argument.

Relative Proj is now done differently (see 18.2). I am now quite happy with the approach, because I have (sadly) given up on dealing with any universal property, as without it, the construction is very easy (when done in the right way). If anyone reads it, please let me know what you think, and tell me what is still confusing. (Summary of feedback to date: people find this an uninspiring topic, but the exercises are gettable.)

In Exercise 18.3.B, we show that the composition of projective morphisms is projective if the final target is quasicompact. (That wacky hypothesis is part of the sign that the notion of projective notion is not great.) I am curious: does anyone know a counterexample without the quasicompactness hypothesis? This isn’t important (it will undoubtedly never come up for me in real life). [Update August 21, 2012: I’ve now asked it on mathoverflow.]

(Update March 27, 2012: there were many typos in the Chow’s Lemma section, so a revised version is now here.) In 20.8, I prove the following form of Chow’s Lemma: if $\pi: X \rightarrow \text{Spec} A$ is proper, and $A$ is Noetherian, then there exists $\rho: X' \rightarrow X$ surjective and projective, with $\pi \circ \rho$ also projective, and with $\rho$ an isomorphism on a dense open subset of $X$. I want to include all other versions that reasonable people (or even reasonably unreasonable people) might reasonably use — with references, but most likely without proofs. The versions I can think of are: (i) weaken “proper” to “finite type and separated”, and weaken the conclusion to “$\pi \circ \rho$ is quasiprojective” (rather than projective), and (ii) a generalization where $\text{Spec} A$ is replaced by a Noetherian scheme, and (iii) = (ii)+(i) (EGA II.5.6.1). If $X$ is reduced, or irreducible, or integral, then we can obviously take $X'$ to be as well. EGA II.5.6 has a variant where the target is quasicompact and separated, with a finite number of irreducible components. Are there any other variants I should care about?